Problem Statement
A computer server monitors incoming connection attempts during 100 separate one-minute intervals. The observed frequencies are:\n\nAttempts: $0$ ($18$), $1$ ($28$), $2$ ($26$), $3$ ($16$), $4$ ($8$), $\ge 5$ ($4$).
**(a)** [2 marks] Show that the sample mean rate of connection attempts per minute is $\bar{x} = 1.80$.
**(b)** [4 marks] Assuming a Poisson model $X \sim \text{Po}(1.8)$, calculate the expected frequency for $X = 0, 1, 2$ and pool cells if necessary.
**(c)** [3 marks] Given a calculated test statistic $\chi^2_{\text{calc}} = 2.14$ with $3$ degrees of freedom, determine whether the Poisson model is a good fit at the $5\%$ level.
Verified Solution & Marking Scheme
Part (a) [M1]
$\sum f_i x_i = 18(0) + 28(1) + 26(2) + 16(3) + 8(4) + 4(5) = 0 + 28 + 52 + 48 + 32 + 20 = 180$
Part (a) [A1]
$\bar{x} = \frac{180}{100} = 1.80$ (or $1.8$ attempts/min)
Part (b) [M1]
Formula $P(X = x) = \frac{e^{-1.8}(1.8)^x}{x!}$
Part (b) [A1]
$E_0 = 100 \times e^{-1.8} = 16.53$
Part (b) [A1]
$E_1 = 100 \times 1.8 e^{-1.8} = 29.75$
Part (b) [A1]
$E_2 = 100 \times \frac{1.8^2}{2} e^{-1.8} = 26.78$
Part (c) [R1]
Critical value for $\text{df} = 3$ at $\alpha = 0.05$ is $\chi^2_{\text{crit}} = 7.815$
Part (c) [R1]
Since $\chi^2_{\text{calc}} = 2.14 < 7.815$ (or $p$-value $\approx 0.54 > 0.05$)
Part (c) [A1]
Do not reject $H_0$. There is sufficient evidence that the Poisson distribution provides an excellent fit for connection attempts.